p-Fractals and power series–II. Some applications to Hilbert-Kunz theory
نویسندگان
چکیده
We use the results of our paper p-Fractals and power series I—Some 2 variable results (Journal of Algebra 280, 2004, pp. 505–536) to prove the rationality of the Hilbert-Kunz series of a large family of power series, including those of the form ∑ i fi(xi, yi), where the fi(xi, yi) are power series with coefficients in a finite field. The methods are effective, as we illustrate with examples. In the final section, which can be read independently of the others, we obtain more precise results for the Hilbert-Kunz function of the 3 variable power series zD − h(x, y).
منابع مشابه
p-Fractals and Power Series - I Some 2 Variable Results
u1, . . . , ur are in kJx1, . . . , xsK with k and deg(u1, . . . , ur) finite. Intending applications to Hilbert-Kunz theory, we code the numbers deg(u1 1 , . . . , u ar r ) into a function φu, which empirically satisfies many functional equations related to “magnification by p”, where p = chark. p-fractals, introduced here, formalize these ideas. In the first interesting case (r = 3, s = 2), t...
متن کاملTranscendence of Some Hilbert-kunz Multiplicities (modulo a Conjecture)
Suppose that h ∈ F [x, y, z], char F = 2, defines a nodal cubic. In earlier papers we made a precise conjecture as to the Hilbert-Kunz functions attached to the powers of h. Assuming this conjecture we showed that a class of characteristic 2 hypersurfaces has algebraic but not necessarily rational Hilbert-Kunz multiplicities. We now show that if the conjecture holds, then transcendental multipl...
متن کاملiv : 0 90 7 . 24 70 v 1 [ m at h . A C ] 1 5 Ju l 2 00 9 Algebraicity of some Hilbert - Kunz multiplicities ( modulo a conjecture )
Let F be a finite field of characteristic 2 and h be the element x3 + y3 + xyz of F [[x, y, z]]. In an earlier paper we made a precise conjecture as to the values of the colengths of the ideals (x, y, z, h) for q a power of 2. We also showed that if the conjecture holds then the Hilbert-Kunz series of H = uv+ h is algebraic (of degree 2) over Q(w), and that μ(h) is algebraic (explicitly, 4 3+ 5...
متن کاملA Hilbert-kunz Criterion for Solid Closure in Dimension Two (characteristic Zero)
Let I denote a homogeneous R+-primary ideal in a twodimensional normal standard-graded domain over an algebraically closed field of characteristic zero. We show that a homogeneous element f belongs to the solid closure I∗ if and only if eHK(I) = eHK((I, f)), where eHK denotes the (characteristic zero) Hilbert-Kunz multiplicity of an ideal. This provides a version in characteristic zero of the w...
متن کامل2 7 A pr 2 00 5 ZEROS OF { − 1 , 0 , 1 } POWER SERIES AND CONNECTEDNESS LOCI FOR SELF - AFFINE SETS
We consider the set Ω2 of double zeros in (0, 1) for power series with coefficients in {−1, 0, 1}. We prove that Ω2 is disconnected, and estimate minΩ2 with high accuracy. We also show that [2 − η, 1) ⊂ Ω2 for some small, but explicit η > 0 (this was only known for η = 0). These results have applications in the study of infinite Bernoulli convolutions and connectedness properties of self-affine...
متن کامل